A conference table is in the shape of a polygon with sides parallel to either the x-axis or the y-axis. A corner of such a polygon is said to be convex if the internal angle is 90 or concave if the internal angle is 270. If the table has 20 convex corners, then the number of ways in which people can be seated at concave corners is:a)16!b)16c)15!d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?

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A conference table is in the shape of a polygon with sides parallel to either the x-axis or the y-axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the table has 20 convex corners, then the number of ways in which people can be seated at concave corners is:

a)
16!
b)
16
c)
15!
d)
Cannot be determined

Correct answer is option 'C'. Can you explain this answer?

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A conference table is in the shape of a polygon with sides parallel to...

Consider the polygon as shown in the figure. Here 4 corners A, D, G and J are the concave corners and remaining 8 corners are convex corners.
In general, for a polygon with sides parallel to the axes, if n is the number of concave corners and m is the number of convex corners, then we have,

m - n = 4 v

m = 20

n = 16
People can be seated in the concave corners in 15! ways. ...[••• Number of ways of seating people will be calculated as in case of circular seating arrangement] Hence, option 3.

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A conference table is in the shape of a polygon with sides parallel to...

Given:
- The conference table is in the shape of a polygon with sides parallel to either the x-axis or the y-axis.
- A corner of the polygon is convex if the internal angle is 90 degrees.
- A corner of the polygon is concave if the internal angle is 270 degrees.
- The table has 20 convex corners.

To find:
The number of ways in which people can be seated at concave corners.

Solution:
Let's assume that the polygon has n sides. Since the number of convex corners is given as 20, it means that the polygon has 20 internal angles of 90 degrees.

Number of Internal Angles in a Polygon:
The number of internal angles in a polygon can be found using the formula:
Number of Internal Angles = (n - 2) * 180
where n is the number of sides of the polygon.

Number of Concave Corners:
Since the total number of internal angles in the polygon is equal to the sum of the internal angles at convex and concave corners, we can write the equation:
Number of Internal Angles = Number of Convex Corners * 90 + Number of Concave Corners * 270

Substituting the values, we get:
(n - 2) * 180 = 20 * 90 + Number of Concave Corners * 270

Simplifying the equation, we have:
(n - 2) * 180 = 1800 + 270 * Number of Concave Corners

Dividing both sides of the equation by 90, we get:
2 * (n - 2) = 20 + 3 * Number of Concave Corners

Simplifying further, we have:
2n - 4 = 20 + 3 * Number of Concave Corners
2n = 24 + 3 * Number of Concave Corners

Since n represents the number of sides of the polygon, it must be an integer. Looking at the equation, we can observe that the right side of the equation (24 + 3 * Number of Concave Corners) must be divisible by 2 for n to be an integer.

Number of Ways to Seat People:
The number of ways to seat people at concave corners would be equal to the number of ways to arrange the people in the concave corners.

If we have n concave corners, the number of ways to arrange people in these corners would be n!. Therefore, the number of ways to seat people at concave corners is given by the value of n!.

Calculating the Number of Ways:
To calculate the number of ways, we need to find the value of n.

From the equation 2n = 24 + 3 * Number of Concave Corners, we can see that the value of n depends on the value of Number of Concave Corners. However, the value of Number of Concave Corners is not given in the question statement. Hence, we cannot determine the exact value of n and consequently, the number of ways to seat people at concave corners.

Conclusion:
Therefore, the correct answer is option (d) Cannot be determined.

Answered by Wizius Careers · View profile

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A conference table is in the shape of a polygon with sides parallel to either the x-axis or the y-axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the table has 20 convex corners, then the number of ways in which people can be seated at concave corners is:a)16!b)16c)15!d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A conference table is in the shape of a polygon with sides parallel to either the x-axis or the y-axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the table has 20 convex corners, then the number of ways in which people can be seated at concave corners is:a)16!b)16c)15!d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A conference table is in the shape of a polygon with sides parallel to either the x-axis or the y-axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the table has 20 convex corners, then the number of ways in which people can be seated at concave corners is:a)16!b)16c)15!d)Cannot be determinedCorrect answer is option 'C'. Can you explain this answer?.

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